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If xyz is not equal to zero it means neither value, x, y or z can be zero.

1) If we pick values for y and z, -2 and 0,5, then we get 1,5 = 2,5 which obviously doesn't fit. If we take -2 and -0,5 we get 2,5 = 2,5 which fits. In order to fit this y and z must therefore have the same sign. Therefore it can either be negative or positive. Still, whether x(y+z)=0, now depends on the value of x. We can see from the question that if xyz is not equal to zero it means x is also not zero. So, if x can't equal zero, x(y+z), no matter the value of y+z, can not equal zero. Why is it C then, anyone?

Given : xyz not equal to 0 therefore x not equal to 0 , y not equal to 0 and z not equal to 0.

from condition 1 : |y+z| = |y|+|z| therefore both y and z are either positive or negative but not zero(this is given), therefore (y+z) will never be zero. This condtion is not sufficient beacuse if x=0 then x(y+z) becomes '0' else x(y+z) is non zero.

from condition 2 : |x+y| = |x| + |y| therefore x and y are either positive or negative but not zero(this is given) , therefore neither x =0 nor y = 0 . This condtion is not sufficient because what if z= -y then x(y+z) = 0.

From condition 1 and 2 neither x=0 nor y=-z (because they both are either positive or negative). Hence we can say that x(y+z) will never be zero.